Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{q^2 - 36}{q - 6}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = q$ $ b = \sqrt{36} = -6$ So we can rewrite the expression as: $p = \dfrac{({q} {-6})({q} + {6})} {q - 6} $ We can divide the numerator and denominator by $(q - 6)$ on condition that $q \neq 6$ Therefore $p = q + 6; q \neq 6$